Multiplicative cross-correlated noise induced escape rate from a metastable state
Jyotipratim Ray Chaudhuri, Sudip Chattopadhyay, Suman Kumar Banik
aa r X i v : . [ c ond - m a t . s o f t ] F e b Multiplicative cross-correlated noise induced escape rate from a metastable state
Jyotipratim Ray Chaudhuri, ∗ Sudip Chattopadhyay, † and Suman Kumar Banik ‡ Department of Physics, Katwa College, Katwa, Burdwan 713130, India Department of Chemistry, Bengal Engineering and Science University, Shibpur, Howrah 711103, India Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0435, USA (Dated: October 30, 2018)We present an analytical framework to study the escape rate from a metastable state under theinfluence of two external multiplicative cross-correlated noise processes. Starting from a phenomeno-logical stationary Langevin description with multiplicative noise processes, we have investigated theKramers’ theory for activated rate processes in a nonequilibrium open system (one-dimensional innature) driven by two external cross-correlated noise processes which are Gaussian, stationary anddelta correlated. Based on the Fokker-Planck description in phase space, we then derive the escaperate from a metastable state in the moderate to large friction limit to study the effect of degreeof correlation on the same. By employing numerical simulation in the presence of external cross-correlated additive and multiplicative noises we check the validity of our analytical formalism forconstant dissipation, which shows a satisfactory agreement between both the approaches for thespecific choice of noise processes. It is evident both from analytical development and the corre-sponding numerical simulation that the enhancement of rate is possible by increasing the degree ofcorrelation of the external fluctuations.
PACS numbers: 05.40.-a, 02.50.Ey, 82.20.Uv
I. INTRODUCTION
Rate theory deals with the passage of a system fromone stable local minima of the energy landscape to an-other via a potential energy barrier providing relevant in-formation on the long-time behavior of the system withdifferent metastable states and hence is a very usefuland pedagogical model for the microscopic descriptionand understanding of a wide range of physical, chemicaland biological phenomena. Examples include diffusionof atoms in solids or on surfaces, isomerization reactionsin solution, electron transfer processes, and ligand bind-ing in proteins or protein folding reactions. Not onlythat the phenomena of flux transitions in superconduct-ing quantum interference devices can also be modeled byrate theory. Kramers pointed out that the passage ofa system from a metastable state due to thermal agi-tation can be considered as a model Brownian particletrapped in a one-dimensional well representing the reac-tant state which is separated by a barrier of finite heightfrom a deeper well signifying the product state. The par-ticle is supposed to be immersed in a medium such thatthe medium exerts damping force on the particle but atthe same time thermally activates it so that the particlemay gain enough energy from its surroundings to crossthe barrier. Thus, the surrounding fluid provides fric-tion and random noise. Conventionally, the friction isassumed to be Markovian and the noise, Gaussian andwhite. Kramers theory provides a useful framework forthe explanation and prediction of rates and mechanismsfor various barrier crossing phenomena and for chemicalreactions in particular. Most of the above mentioned works are basically dealteither with phenomenological Langevin equation or bymicroscopic system-reservoir formulation with both bi- linear or non-linear coupling scheme. Although, the ef-fect of system-bath linear coupling and associate conse-quences in different branches of molecular sciences is wellstudied, the modeling and the proper microscopic inter-pretation of the nature of non-linear coupling and cor-responding manifestation on the various molecular phe-nomena is a very challenging task posed in front of re-searchers in this field. Tanimura and co-workers ex-plained the phenomena of elastic and inelastic relaxationmechanism and their cross effect in vibrational and Ra-man spectroscopy using non-linear system-bath model.It is now a well documented fact that solvent plays avery crucial role in computing the dynamics of a Brow-nian particle. The dynamics of amodel Brownian particle in an inhomogeneous solventgenerally leads to state dependent diffusion due to theappearance of multiplicative noise and state dependentdissipation.
Although modeling of Brownian mo-tion in presence of inhomogeneous solvent is a nontrivialtask, several approaches have been made to study dif-ferent aspects of molecular sciences, e.g., activated rateprocesses, noise induced transport, stochastic resonance and laser and optics. In the overwhelming majority of the above men-tioned treatments, fluctuations experienced by the sys-tem of interest is of internal origin so that the fluc-tuations and dissipation get related through the cel-ebrated fluctuation-dissipation relation (FDR). How-ever, in some situations, the physical origin of dis-sipation and of fluctuations are different as well asindependent.
Thus, there exists nobalance between the influx (efflux) of energy to (from) thesystem and hence there exist no FDR. As a result of this,such systems (usually termed as nonequilibrium opensystem ) do not approach thermal equilibrium asymp-totically and many interesting phenomena are observedthereof. To the best of our knowledge, creation of an opensystem in a nonequilibrium process can be realized inmany ways. To mention, we briefly discuss about few ofthem in the following which are relevant to the presentwork. An additional perturbation applied to the heatbath can excite few bath modes which effectively leadsto the creation of a nonstationary process where thenonstationarity mainly gets reflected in the dissipationkernel.
Dynamics in this case is mainly guidedby an irreversible process.
An extensive analysis ofthis approach can be found in the very recent work ofPopov and Hernandez. External perturbation on theother hand can directly excite the system mode itself(leaving the bath mode alone) or the local bath modes(leaving the system mode alone). In both the situationsthe additional energy input by an independent sourceleads to a shift in the temperature, thus creating an ef-fective temperature like quantity, keep-ing the underlying dynamical process stationary. An-other immediate effect of the breakdown of FDR in thiscase is the creation of a steady state instead of an equi-librium state in the long time limit. To study the escaperate from a metastable state under the influence of ex-ternal cross-correlated noise processes we have adoptedone of the situations mentioned in the latter case to cre-ate an open system. In the present work we drive thereaction coordinate (the system mode) by multiplicativecross-correlated noise processes, leaving the bath modesas it is. In such a situation, the Brownian particle is en-ergized by an extra input of energy, in addition to thethermal energy provided by the heat bath. This leadsto, what we have tried to study in this paper, an en-hancement of barrier crossing event in the activated rateprocesses within the framework of Markovian stationarydynamics.The barrier crossing dynamics with multiplicative andadditive white noise processes aroused strong interest inthe early eighties, where noise forces that are presentsimultaneously in the dynamical system were usuallytreated as random variables uncorrelated with each other.However, there are situations where fluctuations in somestochastic process may have common origin. If this hap-pens then the statistical properties of the fluctuationsshould not be very much different and can be correlatedto each other. Nonlinear stochastic systems includingnoise terms have drawn interest on a wide scale owingto their numerous applications in the field of molecu-lar sciences. Generally it is observed that under such asituation, the noise affects the dynamics through a sys-tem variable. The cross-correlated noise processes werefirst considered by Fedchenia in the context of hydrody-namics of vortex flow where the author introduced cross-correlation among the fluctuations from a common originthat appear in the time evolution equation of dimension-less modes of flow rates. The interference of additiveand multiplicative white noise processes in the kinetics of the bistable systems was first considered by Fulinskiand Telejko where they mentioned the physical possi-bility of a cross-correlated noise. Madureira et al. havepointed out the probability of cross-correlated noise in arealistic model (ballast resistor) showing bi-stable behav-ior of the system. Recently, Mei et al. have studied theeffects of correlations between additive and multiplica-tive noise on relaxation time in a bi-stable system drivenby cross-correlated noise.It is now well accepted that the effect of correlationbetween additive and multiplicative noise (which is alsotermed as correlated noise processes) is considered indis-pensable in explaining phenomena such as steady stateproperties of a single mode laser, bistable kinetics, stochastic resonance in linear system, steady-state en-tropy production, stochastic resonance and transportof particles, etc. Zhu investigated theoretically thestatistical fluctuations of a single mode laser that in-clude correlations between additive and multiplicativewhite noises and showed that the effect of correlationcan lead to larger intensity fluctuations. One can uti-lize this intensity fluctuations to induce a narrow peakin the probe absorption spectrum as well as significantlymodify the emission spectra of matter strongly resonantwith laser field. As shown by Ai et al. one can use thelogistic differential equation to analyze the effects of en-vironmental fluctuations on cancer cell growth using cor-related Gaussian white noise scheme whereby the degreeof correlation of the noise (environmental intensive fluc-tuations) can cause tumor cell extinction. Berdichevskyand Gitterman showed that the maxima of signal tonoise ratio, as a function of the asymmetry of noise, dis-appears in the absence of the coupling between additiveand multiplicative white noises. Very recently Ghosh etal. have used multiplicative correlated noise processesto study the splitting of Kramers’ rate in a symmetrictriple well potential.As the presence of the cross-correlated noise changesthe Langevin dynamics of the system, it is expectedthat there may exist some additional effect of cross-correlation on the escape rate of a metastable state.Study of this additional effect can perfectly serve as amotivation of our work presented in this paper. Toachieve this we extend our recently developed theoreti-cal approach to study the effect of multiplicative cross-correlated noise on the escape rate from a metastablestate. In this present study, the external fluctuation ap-plied to the system under consideration is assumed tobe independent of the system’s characteristic dissipation.In this article, we study the reaction rate (in one di-mension) under the influence of both the internal (ther-mal) noise and external (non-thermal) cross-correlatednoises simultaneously to examine the role of correlationbetween external delta correlated fluctuations on the es-cape rate from a metastable state when the dissipation isspace dependent and the noises appear multiplicativelyin the dynamical equation. In our model the systemis externally driven by two cross-correlated fluctuations.Starting from phenomenological Langevin equation withspace dependent dissipation and multiplicative noises, weconstruct the corresponding Fokker-Planck equation inphase space. Under proper boundary conditions, we solvethe Fokker-Planck equation to calculate the escape ratefor moderate to large dissipation. In the numerical im-plementation of our development the steady state rateexpression (derived analytically) is checked with stochas-tic simulation to establish the fact that the incorporationof external perturbation through cross-correlated fluctu-ations in to the traditional Kramers’ model enhances therate across the barrier top.The organization of the paper is as follows. In sectionII, starting from a phenomenological Langevin equationfor an open system nonlinearly coupled with the envi-ronment and simultaneously acted upon by two cross-correlated multiplicative white noise processes, we con-struct the Fokker-Planck description of the underlyingstochastic process which is multiplicative in general. Wethen calculate the escape rate from a metastable state toexamine the barrier crossing dynamics. A typical exam-ple has been considered in section III to study the effect ofthe cross-correlated fluctuations on the escape rate. Thepaper is then concluded in section IV. To make the paperself contained we provide the derivation of the Fokker-Planck equation and the calculation of escape rate in thetwo appendices. II. CORRELATED NOISE INDUCED ESCAPEFROM A METASTABLE STATE
We consider the motion of a particle of unit mass mov-ing in a Kramers type potential V ( x ) such that it is actedupon by random force f ( t ) of internal origin, i.e., f ( t )originates due to the coupling of the system with its en-vironment and hence is connected to the friction throughthe fluctuation-dissipation relation. Apart from the in-ternal noise, we assume that the system is acted uponby two external Gaussian noise processes, ǫ ( t ) and π ( t ),both of which have a common origin and consequentlyare correlated. Thus, from the very mode of descriptionof our model, it is evident that the system is open innature as it is driven externally by two correlated fluctu-ations. The dynamics of the particle is governed by theLangevin equation¨ x = − Γ( x ) ˙ x − V ′ ( x ) + h ( x ) f ( t ) + g ( x ) ǫ ( t ) + g ( x ) π ( t ) , (1)with h ( x ) = p k B T Γ( x ) , (2)where Γ( x ) is the space dependent friction that arises dueto non-linear system-reservoir coupling. Here, g ( x )and g ( x ) are two arbitrary smooth functions of x andtheir presence makes the external noise processes multi-plicative. T is the thermal equilibrium temperature and k B is the Boltzmann constant. ǫ ( t ) and π ( t ) are Gaussianwhite noise processes with statistical properties h ǫ ( t ) i = h π ( t ) i = 0 (3a) h ǫ ( t ) ǫ ( t ′ ) i = 2 D ǫ δ ( t − t ′ ) (3b) h π ( t ) π ( t ′ ) i = 2 D π δ ( t − t ′ ) (3c) h ǫ ( t ) π ( t ′ ) i = h π ( t ) ǫ ( t ′ ) i = 2 λ p D ǫ D π δ ( t − t ′ ) . (3d)In the above equations (3b-3d), D ǫ and D π are thestrength of the fluctuations ǫ ( t ) and π ( t ), respectivelyand λ (0 λ <
1) denotes the degree of correlationbetween the noise processes ǫ ( t ) and π ( t ). The internalnoise f ( t ) is also assumed to be Gaussian and delta cor-related with statistical properties h f ( t ) i = 0 and h f ( t ) f ( t ′ ) i = 2 δ ( t − t ′ ) . (4)In the above equations h· · · i implies the average overthe realizations of the noise (external or internal) pro-cesses. Eq.(2) along with the second relation of Eq.(4)comprises the fluctuation-dissipation which relates thedamping function Γ( x ) with the fluctuation f ( t ). Theexternal noise processes ǫ ( t ) and π ( t ) are independent ofthe dissipation function and there is no correspondingfluctuation-dissipation relation. We further assume that f ( t ) is independent of ǫ ( t ) and π ( t ) so that h f ( t ) ǫ ( t ′ ) i = h f ( t ) π ( t ′ ) i = 0 . (5)In the absence of the external noise processes, thesystem being closed, the fluctuation-dissipation relationeventually brings it to a stationary state and conse-quently one can examine the barrier dynamics of thesystem using standard methods applicable for a ther-modynamically closed system. The external fluctuationsand their correlation modify the dynamics of activationin two ways. First, they influence the dynamics in theregion around the barrier top so that the effective sta-tionary flux across it gets modified. Second, in the pres-ence of these fluctuations, the equilibrium distributionof the source well is disturbed so that one has to con-sider a new stationary distribution, if any, instead of thestandard equilibrium Boltzmann distribution. This newstationary distribution must be a solution of the Fokker-Planck equation around the bottom of the source wellregion and serves as an appropriate boundary conditionanalogous to Kramers problem.Keeping this in mind we write the correspondingFokker-Planck equation in the phase space [i.e. in ( x, v )space of the system, where v = ˙ x ≡ dx/dt ], describing thedynamics of the Langevin equation (1) [see Appendix-Afor detailed derivation] ∂P∂t = − v ∂P∂x + [Γ( x ) v + V ′ ( x )] ∂P∂v + A ( x ) ∂ P∂v +Γ( x ) P, (6)where A ( x ) = k B T Γ( x ) + g ( x ) (7) g ( x ) = n D ǫ g ( x ) + 2 λ p D ǫ D π g ( x ) g ( x )+ D π g ( x ) (cid:9) / . (8)The diffusion coefficient A ( x ) in the Fokker-Planck equa-tion (6) is an implicit function of the correlation param-eter λ (see the expression of g ( x )). Thus by increasingthe value of λ one can increase the value of the diffu-sion coefficient. This property gets directly reflected inthe final rate expression and in the expression of effectivetemperature (see Eqs.(12) and (13)). In deriving Eq.(6),we have made use of Eqs.(3a-3d) and (4) and the factthat f ( t ) is independent of ǫ ( t ) and π ( t ). It should benoted that when the noise is purely internal and for linearsystem-reservoir coupling, Eq.(6) reduces to the Kramersequation. Kramers’ model for a chemical reaction consists of aparticle undergoing Brownian motion whose coordinate x corresponds to the reaction coordinate and v = dx/dt the reaction rate. In addition to that in Kramers’ originaltreatment, the dynamics of the Brownian particle wasgoverned by Markovian random process. Since the workof Kramers, a number of workers have extended Kramersmodel for non-Markovian case and for state dependentdiffusion to derive the expression for the escape rate. Inorder to allow ourselves a comparison with the Fokker-Planck equation of other forms, we note that though theunderlying dynamics is Markovian, the diffusion coeffi-cient in Eq.(8) is coordinate dependent. It is customaryto get rid of this dependence by approximating the co-efficients at the barrier top or potential well where weneed the steady-state solution of Eq.(8). One may alsouse mean field solution of Eq.(8) obtained by neglectingthe fluctuation terms and putting approximate station-ary condition in the diffusion coefficient.For harmonic oscillator with frequency ω , V ( x ) ≈ E + ω ( x − x ) /
2, the linearized version of the Fokker-Planck equation can be represented as ∂P∂t = − v ∂P∂x + Γ( x ) P + [Γ( x ) v + ω ( x − x )] ∂P∂v + A ∂ P∂v , (9)where A = k B T Γ( x ) + g ( x ), is calculated at the bot-tom of the potential ( x ≈ x ). The general steady statesolution of Eq.(9) becomes P st ( x, v ) = 1 Z exp (cid:18) − v D − ω ( x − x ) D (cid:19) , (10)with D = A / Γ( x ) and Z is the normalization con-stant. The solution (10) can be verified by direct substi-tution in the steady state version of the Fokker-Planck equation (9), namely − v ∂P st ∂x + Γ( x ) P st + [Γ( x ) v + ω ( x − x )] ∂P st ∂v + A ∂ P st ∂v = 0 . It should be noted that the distribution (10) is not anusual equilibrium distribution, rather it serves as sta-tionary distribution for the non-equilibrium open system.This stationary distribution plays the role of an equilib-rium distribution of the closed system which can, how-ever, be recovered in the absence of the external noise.We now embark on the problem of decay of a metastable state in the presence of external cross-correlatednoise processes. In Kramers’ approach, the particle co-ordinate x corresponds to the reaction coordinate, andits values at the minima of the potential well V ( x ) areseparated by a potential barrier to describe the reactantand product states. Linearizing the motion around thebarrier top at x ≈ x b , the steady state version of theFokker-Planck equation corresponding to Eq.(6) reads as − v ∂P stb ∂x − ω b ( x − x b ) ∂P stb ∂v + Γ( x b ) ∂ ( vP stb ) ∂v + A b ∂ P stb ∂v = 0 , (11)where, V ( x ) ≈ E b − ω b ( x − x b ) / ω b >
0, and thesuffix ‘ b ’ indicates that coefficients are to be calculatedusing the general definition of A at the barrier top.After imposing appropriate physically motivatedboundary conditions we then derive the escape rate fornonequilibrium open systems valid in the moderate tostrong friction limit [see Appendix-B for detailed calcu-lation] k = ω π D b √ D r Λ1 + Λ D b exp (cid:18) − ED b (cid:19) , (12)where E = E b − E is the potential barrier height, and D b = A b / Γ( x b ). Already we have mentioned in the in-troduction, open system mean the system is not ther-modynamically closed as it is driven by two noises ofexternal origin and consequently there is no fluctuation-dissipation relation which is mainly responsible to estab-lish the thermal equilibrium (for closed system). It isalso important to note that the escape from a metastablestate intrinsically a non-equilibrium phenomenon as fluc-tuations establish a steady mass motion to give rise to anon-vanishing constant current over the potential barrier.In the steady state rate expression (12), all informationabout the thermal temperature due to internal noise pro-cesses, the strength of both the external noises and theposition dependent dissipation, Γ( x ), are hidden in thequantities D , D b and Λ. Furthermore, the rate constant k is an implicit function of the degree of correlation, λ ,between the external noise processes. We conclude thissection by mentioning the fact that the λ -dependenceof the effective temperature ( D b ) of our model does notdepend on the detailed forms of the coupling functions g ( x ) and g ( x ). III. ANALYSIS OF THE INTERFERENCEBETWEEN TWO EXTERNAL FLUCTUATIONS
In this section we present the numerical implementa-tion of our presently developed theory to establish itspotentiality and applicability to demonstrate the effectof correlation of two external noises ǫ ( t ) and π ( t ) on therate. Eq.(12) is the theoretical expression for the escaperate and is applicable for any two functions g ( x ) and g ( x ) appeared in equation (1). To test the validity ofthe rate expression (12), one needs to assume the par-ticular forms of g ( x ) and g ( x ). In addition to thatan explicit form of the damping term Γ( x ) is needed.For simplicity we use a constant dissipation Γ( x ) = γ inour numerical simulation. Thus the explicit expressionfor the escape rate for constant dissipation and arbitrary g ( x ) and g ( x ) becomes k = ω πω b (cid:20) γk B T + g ( x b ) γk B T + g ( x ) (cid:21) / "(cid:18) γ + ω b (cid:19) / − γ × exp (cid:18) − γEγk B T + g ( x b ) (cid:19) , (13)where the quantities g ( x ) and g ( x b ) are evaluatedaround the potential minima ( x ≈ x ) and maxima( x ≈ x b ), respectively, using the function g ( x ) definedin Eq.(8). In the above equation the quantity g ( x b ) /γ in the exponential factor together with the thermal en-ergy k B T constitutes the effective temperature, a typicalsignature of the open system. Another ef-fect of space dependent diffusion in the dynamics is thepresence of reaction coordinate ( x and x b ) in the steadystate rate expression (13) which is typically absent inthe standard Kramers’ expression (see Eq.(14) below).For D ǫ = 0 and D π = 0, i.e., in the absence of exter-nal driving, Eq.(13) yields Kramers’ rate expression forpure thermal fluctuations valid in the moderate to strongdamping limit k Kramers = ω πω b "(cid:18) γ + ω b (cid:19) / − γ × exp (cid:18) − Ek B T (cid:19) , (14)which can be also verified easily using the explicit formsof the parameters D , D b and Λ in Eq.(12) in the limit ǫ ( t ) = π ( t ) = 0. It is thus clear that in the absence of theexternal noise processes, the derived rate expression (13)reduces to standard Kramers’ escape rate in the moderateto large damping regime.To study the dynamics, we consider a model cubic po-tential of the form V ( x ) = b x − b x where, b and k γ λ = 0.0 λ = 0.5 λ = 0.9 FIG. 1: Plot of barrier crossing rate k as a function of dissi-pation constant γ for various values of correlation parameter λ and for g ( x ) = x and g ( x ) = 1. The solid lines are drawnfrom the theoretical expression, Eq.(13) and the symbols arethe results of numerical simulation of Eq.(1). The values ofthe parameters used are k B T = 0 . D ǫ = D π = 0 . λ =0, 0.5 and 0.9. b are the two constant parameters with b , b >
0, sothat the potential barrier height becomes 4 b / b and x b = 2 b / b . In our numerical simulation we have used b = b = 1. While numerically solving the Langevinequation (1) for constant dissipation, γ , we used a specificcombination of the multiplicative terms, e.g., g ( x ) = x and g ( x ) = 1. Although both the external noise pro-cesses in our model are multiplicative in nature, the spe-cific choice of the external noise processes we adopt inthe numerical simulation captures the essential featureof our model (see the discussion in the following para-graph). We then numerically solve the Langevin equa-tion (1) by employing stochastic simulation algorithm. The numerical rate has been defined as the inverse of themean first passage time and has been calculatedby averaging over 10,000 trajectories. To ensure the sta-bility of our simulation, we have used a small integrationtime step ∆ t = 0 . k is plotted as a function of the dissipation constant, γ , in the moderate to large damping regime where ourtheory is valid, for various values of the degree of corre-lation λ and observe that for a given γ , the escape rate k increases with an increase in λ . The result shows areasonable agreement between the theory and the sim-ulation. It is easy to check that for a particular com-bination of g ( x ) and g ( x ), by increasing the correla-tion parameter λ one basically increases the value of thefunction g ( x ) (see Eq.(8)) evaluated at x ≈ x b . This in-crement in g ( x b ) increases the effective temperature likequantity k B T + ( g ( x b ) /γ ) for a fixed value of γ which inturn pumps more energy into the system which eventu-ally helps in crossing the barrier and increases the escaperate. Another way to explain this phenomenon is to lookat the diffusion coefficient, A b across the barrier. Fromthe expression of A ( x ) (see Eq.(7)) evaluated at x ≈ x b ,i.e., A b = γk B T + g ( x b ) it is clear that g ( x b ) increaseswith an increase in λ for a particular choice of g ( x ) and g ( x ) and for a fixed value of γk B T . This increment inthe diffusion across the barrier enhances the reaction rateas observed. This is the central result of this paper. IV. CONCLUSIONS
In conclusion, we have extended our recently devel-oped theoretical approach of studying escape rate froma metastable state under the influence of external addi-tive cross-correlated noise processes to investigate theeffect of multiplicative noise. In contrast to our previousapproach of using effective correlated noise constructedfrom two additive colored noise processes (see Eq.(1) ofRef. 38), the correlated noise used in the present work hasbeen constructed from multiplicative white noises. Themultiplicative nature of the correlated noises introduces aspace dependent diffusion in the resultant Fokker-Planckequation (6) and modifies the exponential as well as thepre-exponential factor of the steady state rate expression(13). To check the validity of the steady state analyt-ical rate expression, we have performed numerical sim-ulation of the starting Langevin equation (1) with onemultiplicative noise and the other additive one, whichshows a satisfactory agreement between the theory andthe numerics. The analytical development as well as thecorresponding numerical simulation lead us to concludethat the enhancement of rate is possible by increasing thedegree of correlation of the external fluctuations. So far,we have used the formalism of Markovian stochastic pro-cesses in this paper which can be extended to look intothe dynamics within the framework of non-Markovianformalism. We plan to address this issue in our futurecommunication. Acknowledgments
JRC and SC would like to acknowledge the UGC, Indiafor funding through the schemes PSW-103/06-07 (ERO)and 32-304/2006 (SR). SKB acknowledges support fromDepartment of Physics, Virginia Tech.
APPENDIX A: DERIVATION OF THEFOKKER-PLANCK EQUATION
In this appendix we give the detailed calculation ofconstructing the Fokker-Planck equation (6) with spacedependent diffusion corresponding to equation (1) which can be written as:˙ x = v, ˙ v = − Γ( x ) v − V ′ ( x ) + h ( x ) f ( t ) (A1)+ g ( x ) ǫ ( t ) + g ( x ) π ( t ) , We then rewrite the above equation in the following form:˙ u = F ( u , u , t ; f ( t ) , ǫ ( t ) , π ( t )) , ˙ u = F ( u , u , t ; f ( t ) , ǫ ( t ) , π ( t )) , (A2)where we use the following abbreviations u = x, u = v, (A3) F = v, F = − Γ( x ) v − V ′ ( x ) + h ( x ) f ( t ) + g ( x ) ǫ ( t )+ g ( x ) π ( t ) . (A4)The vector u with components u and u thus representsa point in a two-dimensional ‘phase space’ and equation(A2) determines the velocity at each point in the phasespace. The conservation of the phase points now assertsthe following linear equations of motion for density ρ ( u, t )in phase space . ∂∂t ρ ( u, t ) = − X n =1 ∂∂u n F n ( u, t, f ( t ) , ǫ ( t ) , π ( t )) ρ ( u, t ) , (A5)or more compactly ∂ρ∂t = −∇ · F ρ. (A6)Our next task is to find out a differential equation whoseaverage solution is given by h ρ i , where the stochas-tic averaging has to be performed over both the inter-nal noise process f ( t ) and the external noise processes ǫ ( t ) and π ( t ). To this end we note that ∇ · F can bepartitioned into two parts: a constant part ∇ · F and afluctuating part ∇ · F ( t ) containing these fluctuations.Thus, we write ∇ · F ( u, t, f ( t ) , ǫ ( t ) , π ( t ))= ∇ · F ( u ) + β ∇ · F ( u, t, f ( t ) , ǫ ( t ) , π ( t )) , where β is a parameter (we put it as an external param-eter to keep track of the perturbation equation; we put β = 1 at the end of the calculation) and also note that h F ( t ) i = 0. Equation (A6) therefore takes the followingform: ˙ ρ ( u, t ) = ( A + βA ) ρ ( u, t ) , (A7)where A = −∇ · F and A = ∇ · F . The symbol ∇ isused for the operator that differentiates everything thatcomes after it with respect to u . Making use of one of themain results for the theory of linear equation of the form(A7) with multiplicative noise, we derive an averageequation for ρ [ h ρ i = P ( u, t ) the probability density of u ( t )], ∂P∂t = (cid:26) A + β Z ∞ dτ h A ( t ) exp( τ A ) A ( t − τ ) i× exp ( − τ A ) } P. (A8)Equation (A8) is exact when the correlation times of fluc-tuations tend to zero. Using the expressions for A and A we obtain ∂P∂t = (cid:26) −∇ · F + β Z ∞ dτ h∇ · F ( t ) × exp( − τ ∇ · F ) ∇ · F ( t − τ ) i exp( τ ∇ · F ) } P. (A9)The operator exp( τ ∇· F ) in the above equation providesthe solution to the equation ∂y ( u, t ) ∂t = −∇ · F y ( u, t ) , (A10)( y signifies the unperturbed part of ρ ), which can befound explicitly in terms of characteristic curves. Theequation ˙ u = F ( u ) , (A11)for fixed t determines a mapping from u ( τ = 0) to u ( τ ) i.e. , u → u τ with the inverse ( u τ ) − τ = u . The solutionof equation (A10) is y ( u, t ) = y ( u − t , (cid:12)(cid:12)(cid:12)(cid:12) d ( u − t ) d ( u ) (cid:12)(cid:12)(cid:12)(cid:12) = exp( − t ∇ · F ) y ( u, , (A12) | d ( u − t ) /d ( u ) | being a Jacobian determinant. The effectof exp( − t ∇ · F ) on y ( u ) is as follows:exp( − t ∇ · F ) y ( u,
0) = y ( u − t , (cid:12)(cid:12)(cid:12)(cid:12) d ( u − t ) d ( u ) (cid:12)(cid:12)(cid:12)(cid:12) . (A13)The above simplification when put in equation (A9)yields ∂P ( u, t ) ∂t = ∇ · (cid:26) − F + β Z ∞ dτ (cid:12)(cid:12)(cid:12)(cid:12) d ( u − τ ) d ( u ) (cid:12)(cid:12)(cid:12)(cid:12) × (cid:10) F ( u, t ) ∇ − τ · F ( u − τ , t − τ ) (cid:11) × (cid:12)(cid:12)(cid:12)(cid:12) d ( u ) d ( u − τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) P ( u, t ) . (A14)where ∇ − τ denotes differentiation with respect to u − τ .We put β = 1 for the rest of the treatment. Identifying u = x and u = v we now write F = v, F = 0 ,F = − Γ( x ) v − V ′ ( x ) ,F = h ( x ) f ( t ) + g ( x ) ǫ ( t ) + g ( x ) π ( t ) . (A15) In this situation, equation (A14) now reduces to ∂P∂t = − ∂∂x ( vP ) + ∂∂v { Γ( x ) v + V ′ ( x ) } P + ∂∂v Z ∞ dτ h [ h ( x ) f ( t ) + g ( x ) ǫ ( t ) + g ( x ) π ( t )+ ∂∂v − τ { h ( x − τ ) f ( t − τ ) + g ( x − τ ) ǫ ( t − τ )+ g ( x − τ ) π ( t − τ )) } ] (cid:11) P, (A16)where we have used the fact that the Jacobian obeys theequation ddt log (cid:12)(cid:12)(cid:12)(cid:12) d ( x t , v t ) d ( x, v ) (cid:12)(cid:12)(cid:12)(cid:12) = ∂v∂x + ∂∂v {− Γ v + V ′ ( x ) } = − Γ , so that the Jacobian becomes exp( − Γ( x ) t ). Now neglect-ing the terms O ( τ ) we may have x − τ = x − τ v and v − τ = v + Γ τ v + τ V ( x ). The above two equations yield ∂∂v − τ = (1 − Γ τ ) ∂∂v + τ ∂∂x . (A17)Taking this in to consideration of equation (A17), equa-tion (A16) can be simplified in the following form: ∂P∂t = − ∂ ( vP ) ∂x + ∂∂v [Γ( x ) v + V ′ ( x )] P + ∂ ∂v [ k B T Γ( x ) + g ( x )] P, (A18)where g ( x ) = n D ǫ g ( x ) + 2 λ p D ǫ D π g ( x ) g ( x )+ D π g ( x ) (cid:9) / . (A19)Defining A ( x ) = k B T Γ( x ) + g ( x ) , (A20)the above equation (A18) can be written as ∂P∂t = − v ∂P∂x + [Γ( x ) v + V ′ ( x )] ∂P∂v + A ( x ) ∂ P∂v +Γ( x ) P, (A21)which is our required Fokker-Planck equation. APPENDIX B: CALCULATION OF THE ESCAPERATE
In this appendix we show the detailed calculationof the escape rate for nonequilibrium open system.The technique we adopt here resembles our previousapproaches but makes the current paper self con-tained. Following Kramers, we make ansatz that the non-equilibrium steady state probability P stb ( x, v ) generat-ing a non-vanishing diffusion current across the barrieris given by P stb ( x, v ) = exp (cid:18) − v D b − V ( x ) D b (cid:19) G ( x, v ) , (B1)where D b = A b / Γ( x b ). Inserting Eq.(B1) in (11), weobtain the equation for G ( x, v ) using the steady state inthe neighborhood of x b − v ∂G∂x − [ ω b ( x − x b ) + Γ( x ) v ] ∂G∂v + A b ∂ G∂v = 0 . (B2)At this point we set y = v + a ( x − x b ) , (B3)where a is a constant to be determined. With the helpof the transformation (B3), Eq.(B2) reduces to A b d Gdy − [ ω b ( x − x b ) + { Γ( x b ) + a } v ] dGdy = 0 . (B4)Now let [ ω b ( x − x b ) + { Γ( x b ) + a } v ] = − µy, (B5)where µ is another constant. By virtue of the relation(B5), Eq.(B4) becomes d Gdy + Λ y dGdy = 0 , (B6)where Λ = µ/A b with A b = k B T Γ( x b ) + g ( x b ). Theconstant µ and a must satisfy the following equationssimultaneously: − µa = ω b and − µ = Γ( x b ) + a. (B7)This implies that the constant a must satisfy the follow-ing quadratic equation a + Γ( x b ) a − ω b = 0 , which allows the solution for a as a ± = 12 (cid:26) − Γ( x b ) ± q Γ ( x b ) + 4 ω b (cid:27) . (B8)Thus, the general solution of (B6) is G ( y ) = G Z y exp (cid:18) − Λ z (cid:19) dz + G , (B9)where G and G are two constants of integration. Welook for a solution which vanishes for large x . This con-dition is satisfied if the integration in (B9) remains finitefor | y | → + ∞ . This implies that Λ > a − becomes relevant. Then the requirement P b ( x, v ) → x → + ∞ yields G = G r π . (B10) Thus we have G ( y ) = G (cid:20)r π
2Λ + Z y exp (cid:18) − Λ z (cid:19) dz (cid:21) , (B11)and correspondingly, P stb ( x, v ) = G (cid:20)r π
2Λ + Z y exp (cid:18) − Λ z (cid:19) dz (cid:21) × exp (cid:18) − v D b − V ( x ) D b (cid:19) . (B12)The current across the barrier associated with this steadystate distribution is given by j = Z + ∞−∞ vP stb ( x ≈ x b , v ) dv, which may be evaluated using (B12) and the linearizedversion of V ( x ), namely V ( x ) ≈ E b − ω b ( x − x b ) / j = G s π Λ + D − b D b exp (cid:18) − E b D b (cid:19) . (B13)To determine the remaining constant G we note thatas x → −∞ , the pre-exponential factor in (B12) reducesto G p π/ Λ. We then obtain the reduced distributionfunction in x as˜ P stb ( x → −∞ ) = 2 πG r D b Λ exp (cid:18) − V ( x ) D b (cid:19) , (B14)where we have used the definition for the reduceddistribution as˜ P ( x ) = Z −∞ + ∞ P ( x, v ) dv. Similarly, we derive the reduced distribution in the leftwell around x ≈ x using Eq.(10) where the linearizedpotential is V ( x ) ≈ E + ω ( x − x ) / P st ( x ) = 1 Z p πD exp (cid:18) − ω ( x − x ) D (cid:19) , (B15)with the normalization constant given by 1 /Z = ω / (2 πD ) The comparison of the distribution (B14) and(B15) near x ≈ x , gives, G = r Λ D b ω π √ πD . (B16)Hence, from (B13), the normalized current or the barriercrossing rate k , for moderate to large friction regime isgiven by k = ω π D b √ D r Λ1 + Λ D b exp (cid:18) − ED b (cid:19) , (B17)where E = E b − E is the potential barrier height. ∗ Electronic address: jprc˙[email protected] † Electronic address: sudip˙chattopadhyay@rediffmail.com ‡ Electronic address: [email protected]; Present Address: De-partment of Biological Sciences, Virginia Polytechnic In-stitute and State University, Blacksburg, VA 24061-0406,USA. H. A. Kramers, Physica , 284 (1940). A. Simon and A. Libchaber, Phys. Rev. Lett. , 3375(1992). E.W.-G. Diau, J. L. Herek, Z. H. Kim, and A. H. Zewail,Science , 847 (1998). L. I. McCann, M. I. Dykman, and B. Golding, Nature ,785 (1999). P. H¨anggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. , 251 (1990). V. I. Mel’nikov, Phys. Rep. , 1 (1991); H. Grabert, P.Schramm, and G. L. Ingold, Phys. Rep. , 115 (1998). U. Weiss,
Quantum Dissipative Systems (World Scientific,Singapore, 1999). A. Nitzan,
Chemical Dynamics in Condensed Phases (Ox-ford University Press, Oxford, 2006). R. F. Grote and J. T. Hynes, J. Chem. Phys. , 2715(1980). E. Pollak, J. Chem. Phys. , 865 (1986). K. Okamura and Y. Tanimura, Phys. Rev. E , 2747(1997); T. Kato and Y. Tanimura, J. Chem. Phys. ,260 (2004). P. Resibois and M. dc Leener,
Chemical Kinetic Theory ofFluids (Wiley-Interscience, New York, 1977). J. R. Chaudhuri, G. Gangopadhyay, and D. S. Ray, J.Chem. Phys. , 5565 (1998). R. Hernandez, J. Chem. Phys. , 7701 (1999); R. Her-nandez and F. L. Somer, J. Phys. Chem. B , 1064(1999). E. Hershkovits and R. Hernandez, J. Chem. Phys. ,014509 (2005). J. M. Moix, T. D. Shepherd, and R. Hernandez, J. Phys.Chem. B , 19476 (2004); J. M. Moix and R. Hernandez,J. Chem. Phys. , 114111 (2005). A. V. Popov and R. Hernandez, J. Chem. Phys. ,244506 (2007). H. W. Hsia, N. Fang, and X. Lee, Phys. Lett. A , 326(1996); A. N. Drozdov and S. C. Tucker, J. Phys. Chem.B , 6675 (2001). J. R. Chaudhuri, D. Barik, and S.K. Banik, J. Phys. A ,14715 (2007). J. R. Chaudhuri, S. Chattopadhyay, and S.K. Banik, J.Chem. Phys. , 224508 (2007). J. R. Chaudhuri, D. Barik, and S. K. Banik, Phys. Rev. E , 061119 (2006). E. Pollak and P. Talkner, Chaos , 026116 (2005) andreferences therein. E. Pollak and A. M. Berezhkovskii, J. Chem. Phys. ,1344 (1993). Y. M. Blanter and M. B¨uttiker, Phys. Rev. Lett. , 4040(1998); Phys. Rep. , 1 (2000); R. Krishnan, M. C. Ma-hato, and A. M. Jayannavar, Phys. Rev. E , 021102(2004). M. O. Magnasco, Phys. Rev. Lett. , 1477 (1993). P. Reimann, M. Grifoni, and P. H¨anggi, Phys. Rev. Lett. , 10 (1997); P. Reimann, Phys. Rep. , 57 (2002). F. Marchesoni, Chem. Phys. Lett. , 20 (1984); A. V.Barzykin and K. Seki, Europhys. Lett. , 117 (1997); Q.Long, L. Cao, Da-jin Wu, and Zai-guang Li, Phys. Lett. A , 339 (1997); O. V. Gerashchenko, S. L. Ginzburg, andM. A. Pustovoit, JETP Lett. , 997 (1998). R. L. Stratonovich,
Topics in the Theory of Random Noise (Gordon and Breach, London, 1967). R. Kubo, M. Toda, N. Hashitsume, and N. Saito,
Sta-tistical Physics II: Nonequilibrium Statistical Mechanics (Springer, Berlin, 1995). W. Horsthemke and R. Lefever,
Noise-induced Transitions (Springer, Berlin, 1994). K. Huang,
Lectures on Statistical Physics and ProteinFolding (World Scientific, Singapore, 2005). J. R. Chaudhuri, S. K. Banik, B. C. Bag, and D. S. Ray,Phys. Rev. E. , 061111 (2001). S. K. Banik, J. Ray Chaudhuri, and D. S. Ray, J. Chem.Phys. , 8330 (2000). J. R. Chaudhuri, D. Barik, and S. K. Banik, Phys. Rev. E , 051101 (2006). J. Mencia Bravo, R. M. Velasco, and J. M. Sancho, J.Math. Phys. , 2023 (1989). K. Lindenberg and B. J. West,
The Nonequilibrium Statis-tical Mechanics of Open and Closed Systems (VCH, NewYork, 1990). M. M. Millonas and C. Ray, Phys. Rev. Lett. , 1110(1995). J. R. Chaudhuri, S. Chattopadhyay, and S. K. Banik, Phys.Rev. E , 021125 (2007). I.I. Fedchenia, J. Stat. Phys. , 1005 (1988). A. Fulinski and T. Telejko, Phys. Letts. A , 11 (1991). A. J. R. Madureira, P. H¨anggi, and H. S. Wio, Phys. Lett.A , 248 (1994). D. Mei, C. Xie, and L. Zhang, Phys. Rev. E , 051102(2003). S. Zhu, Phys. Rev. A , 2405 (1993). Y. Jia and J. R. Li, Phys. Rev. E , 5786 (1996). V. Berdichevsky and M. Gitterman, Phys. Rev. E , 1494(1999). B. C. Bag, S. K. Banik, and D. S. Ray, Phys. Rev. E ,026110 (2001). C. J. Tessone, H. S. Wio, and P. H¨anggi, Phys. Rev. E ,4623 (2000). J. Li, J. Luczka, and P. H¨anggi Phys. Rev. E , 011113(2001). B. Q. Ai, X. J. Wang, G. T. Liu, and L. G. Liu, Phys. Rev.E , 022903 (2003). P. K. Ghosh, B. C. Bag, and D. S. Ray, Phys. Rev. E ,032101 (2007); J. Chem. Phys. , 044510 (2007). P. Jung, Phys. Rep. , 175 (1993); C. Xie, D. Mei, L.Cao, and Da-jin Wu, Eur. Phys. J. B , 83 (2003); P.Majee and B. C. Bag, J. Phys. A , 3353 (2004). J. Garcia-Ojalvo and J. M. Sancho,
Noise in spatially ex-tended system (Springer-Verlag, New York, 1999). J. M. Sancho, M. San Miguel, S. L. Katz, and J. D. Gunton,Phys. Rev. A , 1589 (1982). C. Mahanta and T.G. Venkatesh, Phys. Rev. E , 4141(1998); D. Barik, B. C. Bag, and D. S. Ray, J. Chem. Phys. , 12973 (2003). N. G. van Kampen, Phys. Rep.24